Let a ≠ b be two non-zero real numbers. Then the number of elements in the set X = z ∈ C : Re a z 2 + b z = a and Re b z 2 + a z = b is equal to

Step-by-step solution

Given, X = z ∈ C : Re a z 2 + b z = a   and   Re b z 2 + a z = b Now, let z = x + i y So,   Re a z 2 + b z = a ⇒ Re a x + i y 2 + b x + i y = a ⇒ Re a x 2 - y 2 + 2 i x y + b x + i y = a ⇒ a x 2 - y 2 + b x = a         . . .   i And,   Re b z 2 + a z = b ⇒ Re b x + i y 2 + a x + i y = b ⇒ Re b x 2 - y 2 + 2 i x y + a x + i y = b ⇒   b x 2 - y 2 + a x = b       . . .   ii From i and ii ,   i - ii we g